MINIMAL POLYNOMIALS AND ANNIHILATORS OF GENERALIZED VERMA MODULES OF THE SCALAR TYPE

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1 MINIMAL POLYNOMIALS AND ANNIHILATORS OF GENERALIZED VERMA MODULES OF THE SCALAR TYPE HIROSHI ODA AND TOSHIO OSHIMA Abstract. We construct a generator system of the annihilator of a generalized Verma module of a reductive Lie algebra induced from a character of a parabolic subalgebra as an analogue of the minimal polynomial of a matrix.. Introduction In the representation theory of a real reductive Lie group G the center Zg of the universal enveloping algebra Ug of the complexification g of the Lie algebra of G plays an important role. For example, any irreducible admissible representation τ of G realized in a subspace E of sections of a certain G-homogeneous vector bundle is a simultaneous eigenspace of Zg parameterized by the infinitesimal character of τ. The differential equations induced from Zg are often used to characterize the subspace E. If the representation τ is small, we expect more differential equations corresponding to the primitive ideal I τ, that is, the annihilator of τ in Ug. For the study of I τ and these differential equations it is interesting and important to get a good generator system of I τ. Let p Θ be a parabolic subalgebra containing a Borel subalgebra b of g and let λ be a character of p Θ. Then the generalized Verma module of the scalar type is by definition. M Θ λ = Ug/J Θ λ with J Θ λ = Ug X λx. X p Θ In this paper we construct generator systems of the annihilator Ann M Θ λ of the generalized Verma module M Θ λ in a unified way. If τ can be realized in a space E of sections of a line bundle over a generalized flag manifold, the annihilator of the corresponding generalized Verma module kills E. When g = gl n, [O] and [O3] construct such a generator system by generalized Capelli operators defined through quantized elementary divisors. This is a good generator system and in fact it is used there to characterize the image of the Poisson integrals on various boundaries of the symmetric space and also to define generalized hypergeometric functions. A similar generator system is studied by [Od] for g = o n but it is difficult to construct the corresponding generator system in the case of other general reductive Lie groups. On the other hand, in [O4] we give other generator systems as a quantization of minimal polynomials when g is classical. Associated to a faithful finite dimensional representation π of g and a g-module M, [O4] defines a minimal polynomial q π,m x as is quoted in Definition.3 and Definition.5. If g = gl n and π is a natural representation of g, q π,m x is char- E ij acterized by the condition q π,m F π M =. Here F π = i n j n is the matrix whose i, j-component is the fundamental matrix unit E ij and then F π is identified with a square matrix with components in g Ug. In this case q π,mθλx is naturally regarded as a quantization of the minimal polynomial which corresponds

2 HIROSHI ODA AND TOSHIO OSHIMA to the conjugacy class of matrices given by a classical limit of M Θ λ. For example, if p Θ is a maximal parabolic subalgebra of gl n, the minimal polynomial q π,mθ λx is a polynomial of degree. For general π and g, the matrix F π is the image pe ij of E ij under the contragredient map p of π and then F π is a square matrix of the size dim π with components in g. For example, if π is the natural representation of o n, then the i, j-component of F π equals E ij E ji. In [O4] we calculate the minimal polynomial q π,mθ λx for the natural representation π of each type of classical Lie algebra g and by putting. I π,θ λ = Ugq π,mθ λf π ij + Ug, i,j it is shown that.3 J Θ λ = I π,θ λ + Jλ Θ with Jλ Θ = X b Zg Ann M Θ λ Ug X λx for a generic λ. This equality is essential because it shows that q π,mθ λf π ij give elements killing M Θ λ which cannot be described by Zg and define differential equations characterizing the local sections of the corresponding line bundle of a generalized flag manifold. Moreover.3 assures that I π,θ λ equals Ann M Θ λ for a generic λ Proposition 3.. In this paper, π may be any faithful irreducible finite dimensional representation of a reductive Lie algebra g. In Theorem.4 we calculate a polynomial q π,θ x; λ which is divisible by the the minimal polynomial q π,mθ λx and it is shown in Theorem.9 that the former polynomial equals the latter for a generic λ. If p Θ = b, this result gives the characteristic polynomial associated to π as is stated in Theorem.33, which is studied by [Go]. We prove Theorem.4 in a similar way as in [O4] but in a more generalized way and the proof is used to get the condition for.3. Another proof which is similar as is given in [Go] is also possible and it is based on the decomposition of the tensor product of some finite dimensional representations of g given by Proposition.7. The proof of Theorem.9 uses infinitesimal Mackey s tensor product theorem which is explained in Appendix A. In 3 we examine.3 and obtain a sufficient condition for.3 by Theorem 3.. Proposition 3.5 and Proposition 3.7 assure that a generic λ satisfies this condition if π is one of many proper representations including minuscule representations, adjoint representations, representations of multiplicity free, and representations with regular highest weights. In such cases the sufficient condition is satisfied if λ is not in the union of a certain finite number of complex hypersurfaces in the parameter space, which are defined by the difference of certain weights of the representation π. On the other hand, in Appendix B, we give counter examples for which our sufficient condition is never satisfied by any λ. In Proposition 3.3 we also study the element of Zg contained in I π,θ λ. A corresponding problem in the classical limit is to construct a generator system of the defining ideal of the coadjoint orbit of g and in fact Theorem 3.8 is considered to be the classical limit of Corollary 3.. If π is smaller, the two-sided ideal I π,θ λ is better in general and therefore in 4 we give examples of the characteristic polynomials of some small π for every simple g and describe some minimal polynomials, especially in each case where p Θ is maximal. Note that the minimal polynomial is a divisor of the characteristic polynomial evaluated at the infinitesimal character. In Proposition 4. we present a two-sided ideal of Ug for every g, p Θ and examine the condition.3 for this ideal by applying Theorem 3.. In particular, the condition is satisfied if the infinitesimal character of M Θ λ is regular in the case when g = gl n, o n+, sp n or

3 ANNIHILATORS OF GENERALIZED VERMA MODULES 3 G. The condition is also satisfied if the infinitesimal character is in the positive Weyl chamber containing the infinitesimal characters of the Verma modules which have finite dimensional irreducible quotients. Some applications of our results in this paper to the integral geometry will be found in [O4, 5] and [OSn]. Acknowledgments. The referee gave the authors many helpful comments to make the paper more accessible. The authors greatly appreciate it.. Minimal Polynomials and Characteristic Polynomials For an associative algebra A and a positive integer N, we denote by MN, A the associative algebra of square matrices of size N with components in A. We use the standard notation gl n, o n and sp n for classical Lie algebras over C. The exceptional simple Lie algebra is denoted by its type E 6, E 7, E 8, F 4 or G. The Lie algebra gl N is identified with MN, C EndC N with the bracket [X, Y ] = XY Y X. In general, if we fix a base {v,..., v N } of an N-dimensional vector space V over C, we naturally identify an element X = X ij of MN, C with an element of EndV by Xv j = N i= X ijv i. Let E ij = δ µi δ νj µ N ν N MN, C be the standard matrix units and put E ij = E ji. Note that the symmetric bilinear form. X, Y = Trace XY for X, Y gl N on gl N is non-degenerate and satisfies. E ij, E µν = E ij, E νµ = δ iν δ jµ, X = X, E ji E ij, i,j AdgX, AdgY = X, Y for X, Y gl N and g GLN, C. In general, for a Lie algebra g over C, we denote by Ug and Zg the universal enveloping algebra of g and the center of Ug, respectively. Then we have the following lemma. Lemma. [O4, Lemma.]. Let g be a Lie algebra over C and let π, C N be a representation of g. Let p be a linear map of gl N to Uπg satisfying.3 p[x, Y ] = [X, py ] for X πg and Y gl N, that is, p Hom πg gln, Uπg. Fix qx C[x] and put F = pe ij i N M N, Uπg, j N.4 Q ij = qf M N, Uπg. Then.5 and.6 [X, Q ij ] = i N j N padge ij i N = t g F t g for g GLn, C j N N N X µi Q µj X jν Q iν µ= ν=

4 4 HIROSHI ODA AND TOSHIO OSHIMA = N N X, E iµ Q µj Q iν X, E νj for X = X µν µ= ν= µ N ν N πg. Hence the linear map gl N Uπg defined by E ij Q ij is an element of Hom πg gln, Uπg. In particular, N i= Q ii Zπg. Remark.. The referee suggested that we should give the reader the following conceptual explanation of Lemma.: Since gl N MN, C is naturally identified with MN, C via., the linear πg-homomorphism p : gl N Uπg is considered as an element of gl N Uπg MN, C Uπg MN, Uπg. By this identification, the image of p equals t F and hence.5 holds almost immediately. Also,.6 is equivalent to the fact that MN, C Uπg πg is a subalgebra of MN, C Uπg MN, Uπg. Now we introduce the minimal polynomial defined by [O4], which will be studied in this section. Definition.3 characteristic polynomials and minimal polynomials. Given a Lie algebra g, a faithful finite dimensional representation π, C N and a g-homomorphism p of EndC N gl N to Ug. Here we identify g as a subalgebra of gl N through π. Let Ẑg denote the quotient field of Zg. Recall Zg is an integral domain. Put F = pe ij MN, Ug. We say q F x Ẑg[x] is the characteristic polynomial of F if it is the monic polynomial with the minimal degree which satisfies q F F = in M N, Ẑg Zg Ug. Suppose moreover a g-module M is given. Then we say q F,M x C[x] is the minimal polynomial of the pair F, M if it is the monic polynomial with the minimal degree which satisfies q F,M F M =. Remark.4. The uniqueness of the characteristic or minimal polynomial is clear if it exists. Suppose g is reductive. Then the characteristic polynomial actually exists by [O4, Theorem.6]. The same theorem assures the existence of the minimal polynomial if M has a finite length or an infinitesimal character. Definition.5. If the symmetric bilinear form. is non-degenerate on πg, the orthogonal projection of gl N onto πg satisfies the assumption for p in Lemma., which we call the canonical projection of gl N to πg g. In this case we put F π = pe ij. Then we call q Fπ x resp. q Fπ,M x in Definition.3 the characteristic polynomial of π resp. the minimal polynomial of the pair π, M and denote it by q π x resp. q π,m x. Remark.6. For a given involutive automorphism σ of gl N, put g = {X gl N ; σx = X} and let π be the inclusion map of g gl N. Then px = X+σX. Hereafter in the general theory of minimal polynomials which we shall study, we restrict our attention to a fixed finite dimensional representation π, V of g such that.7 { g is a reductive Lie algebra over C, π is faithful and irreducible. Moreover we put N = dim V and identify V with C N through some basis of V. The assumption of Definition.5 is then satisfied.

5 ANNIHILATORS OF GENERALIZED VERMA MODULES 5 Remark.7. i The dimension of the center of g is at most one. ii Fix g GLV. If we replace π, V by π g, V with π g X = AdgπX for X g in Lemma., F π MN, g is naturally changed into t g F π t g under the fixed identification V C N. This is clear from Lemma. cf. [O4, Remark.7 ii]. iii Exceptionally the condition.7 will not be assumed in Definition.36 and Proposition.37. Definition.8 root system. We fix a Cartan subalgebra a of g and let Σg be a root system for the pair g, a. We choose an order in Σg and denote by Σg + and Ψg the set of the positive roots and the fundamental system, respectively. For each root α Σg we fix a root vector X α g. Let g = n a n be the triangular decomposition of g so that n is spanned by X α with α Σg +. We say µ a is dominant if and only if µ, α α, α / {,,...} for any α Σg+. Let us prepare some lemmas and definitions. Lemma.9. Let U be a k-dimensional subspace of gl N such that, U is nondegenerate. Let p U be the orthogonal projection of gl N to U and let {v,..., v k } be a basis of U with v, v j = for j k. Suppose that u gl N satisfies u, v j = for j k. Then p U u = u,v v,v v. The proof of this lemma is easy and we omit it. Lemma.. Choose a base {v i ; i =,..., N} of V for the identification V C N so that v i are weight vectors with weights ϖ i a, respectively. We identify g with the subalgebra πg of gl N MN, C and put a N = N i= CE ii. For F π = F ij we have.8 i N j N F ii = ϖ i = N ϖ i E jj E jj, j= adhf ij = ϖ i ϖ j HF ij H a, F ij, E µν = with i j implies ϖ i ϖ j = ϖ ν ϖ µ Σg, N a = CF ii a N, n = CF ij, n = i= ϖ i ϖ j Σg + under the identification a a a N a N ϖ j ϖ i Σg + CF ij by the bilinear form.. Proof. Note that H a is identified with N j= ϖ jhe jj a N gl N. Hence adhe ij = ϖ i ϖ j HE ij and therefore adhf ij = ϖ i ϖ j HF ij. In particular we have F ii a. Since N H, F ii = H, E ii = ϖ j HE jj, E ii = ϖ i H j= H a, we get F ii = ϖ i. For each root α, the condition X α ij = X α, E ji = means ϖ i ϖ j = α. Hence if i j and X a+ α Σg, α ϖ j ϖ i CX α, then E ij, X = and therefore F ij, X =. Hence F ij = if i j and ϖ j ϖ i Σg. On the other hand, if ϖ j ϖ i Σg, we can easily get F ij = CX ϖi ϖ j for some C C. Hence F ij, E µν = if ϖ i ϖ j ϖ ν ϖ µ.

6 6 HIROSHI ODA AND TOSHIO OSHIMA Through the identification of a a a N in the lemma, we introduce the symmetric bilinear form, on a. We note this bilinear form is real-valued and positive definite on α Ψg Rα. Now we take a subset Θ Ψg with Θ Ψg and fix it. Definition. generalized Verma module. Put a Θ = {H a; αh =, α Θ}, g Θ = {X g; [X, H] =, H a Θ }, m Θ = {X g Θ ; X, H =, H a Θ }, Σg = {α; α Σg + }, Σg Θ = {α Σg; αh =, H a Θ }, Σg Θ + = Σg Θ Σg +, Σg Θ = { α; α Σg Θ + }, n Θ = CX α, n Θ = CX α, α Σg + \Σg Θ α Σg \Σg Θ b = a + n, p Θ = g Θ + n Θ, ρ = α, ρθ = α, α Σg + α Σg Θ + ρ Θ = ρ ρθ. For Λ a which satisfies Λ,α α,α {,,,...} for α Θ, let U Θ,Λ denote the finite dimensional irreducible g Θ -module with highest weight Λ. By the trivial action of n Θ, we consider U Θ,Λ to be a p Θ -module. Put.9 M Θ,Λ = Ug UpΘ U Θ,Λ. Then M Θ,Λ is called a generalized Verma module of the finite type. Remark.. i p Θ is a parabolic subalgebra containing the Borel subalgebra b. p Θ = m Θ + a Θ + n Θ gives its direct sum decomposition. ii Every finite dimensional irreducible p Θ -module is isomorphic to U Θ,Λ with a suitable choice of Λ. iii M,Λ is nothing but the Verma module for the highest weight Λ a. iv Let u Λ be a highest weight vector of U Θ,Λ. Then u Λ is a highest weight vector of M Θ,Λ. Moreover u Λ generates M Θ,Λ because M Θ,Λ = Ug UpΘ U Θ,Λ = U n Θ C Up Θ UpΘ U Θ,Λ = U n Θ C U Θ,Λ = U n Θ C U n g Θ u Λ = U n u Λ. Hence M Θ,Λ is a highest weight module and is therefore a quotient of the Verma module M,Λ. v If Λ, α = for each α Θ, then dim U Θ,Λ = and we have the character λ Θ of p Θ such that Xu Λ = λ Θ Xu Λ for X p Θ. Since Ug = U n Θ Ug X λθ X X p Θ is a direct sum and M Θ,Λ = U n Θ C Cu Λ, we have the kernel of the surjective Ug-homomorphism Ug M Θ,Λ defined by D D u Λ equals X p Θ Ug X λ Θ X.

7 ANNIHILATORS OF GENERALIZED VERMA MODULES 7 Definition.3 generalized Verma module of the scalar type. For λ a Θ define a character λ Θ of p Θ by λ Θ X + H = λh for X m Θ + n Θ and H a Θ. Put J Θ λ = Ug X λθ X, X p Θ. Jλ Θ = X b Ug X λ Θ X, M Θ λ = Ug/J Θ λ, Mλ Θ = Ug/Jλ Θ. Then M Θ λ is isomorphic to M Θ,λΘ, which is called a generalized Verma module of the scalar type. If Θ =, we denote J λ and M λ by Jλ and Mλ, respectively. Definition.4 Weyl group. Let W denote the Weyl group of Σg, which is generated by the reflections w α : a µ µ µ,α α,α α a with respect to α Ψg. Put. W Θ = {w W ; w Σg + \ Σg Θ = Σg + \ Σg Θ }, W Θ = {w W ; w Σg Θ + Σg + }. Then each element w W Θ is a unique element with the smallest length in the right coset ww Θ and the map W Θ W Θ w, w w w W is a bijection. For w W and µ a, define. w.µ = wµ + ρ ρ. Here we note that W Θ is generated by the reflections w α with α Θ and.3 ρ Θ, α = for α Σg Θ. Definition.5 infinitesimal character. Let D Ug. We denote by D a the element of Ua which satisfies D D a nug + Ugn and identify D a Ua Sa with a polynomial function on a. Then a µ = a w.µ for Zg, µ a, and w W. Let µ a. We say a g-module M has infinitesimal character µ if each Zg operates by the scalar a µ in M. We say an infinitesimal character µ is regular if µ + ρ, α = for any α Σg. Remark.6. The generalized Verma module M Θ,Λ in Definition. has infinitesimal character Λ. It is clear by Remark. iv. Definition.7 Casimir operator. Let {X i ; i =,..., ω} be a basis of g. Then put ω π = X i Xi i= with the dual basis {X i } of {X i} with respect to the symmetric bilinear form. under the identification g gl N through π and call π the Casimir operator of g for π. Remark.8. As is well-known, π Zg and π does not depend on the choice of {X i }. We may assume in Definition.7 that {X,..., X ω } and {X ω +,..., X ω } be bases of g Θ and n Θ + n Θ, respectively. Then Xi g Θ for i =,..., ω and.4 Θ π = X i Xi is the Casimir operator of g Θ for π. ω i=

8 8 HIROSHI ODA AND TOSHIO OSHIMA Lemma.9. Fix a basis {H,..., H r } of the Cartan subalgebra a of g. i Let {H,..., H r } be the dual basis of {H,..., H r }. Put H α = [X α, X α ]. Then π = = α Σg X α X α r X α, X α + H i Hi r H i Hi + i= = Θ π + α Σg + α Σg + \Σg Θ i= X α X α X α, X α + αh αh α H α, H α Xα X α X α, X α αh αh α. H α, H α ii Let M be a highest weight module of g with highest weight µ a. Then π v = µ, µ + ρ v for any v M. iii Let v be a weight vector of π belonging to an irreducible representation of g Θ realized as a subrepresentation of π gθ and let ϖ denote the lowest weight of the irreducible subrepresentation. Then α Σg + \Σg Θ π v = π, π ρ v, Θ π v = ϖ, ϖ ρθ v, X α X α X α, X α v = π ϖ, π + ϖ ρ v. Here π denotes the lowest weight of π. iv Fix β Σg + and put gβ = CX β + CX β + r i= CH i. Let v be a weight vector of π belonging to an irreducible representation of gβ realized as a subrepresentation of π gβ and let ϖ denote the lowest weight of the irreducible subrepresentation. Let ϖ + lβ be the weight of v. Then X β X β.5 X β, X β v = l ϖ + l β, β v. v Suppose g is simple. Let α max is the maximal root of Σg + and let B, be the Killing form of g. Then Proof. i Note that Bα max, α max + ρ =..6 H α, H α = H α, [X α, X α ] = [H α, X α ], X α = αh α X α, X α Since the dual base of {X α, H i ; α Σg, i =,..., r} equals { X α,x α, H i ; α Σg, i =,..., r}, the claim is clear. ii Let v µ be a highest weight vector of M. Then r π v µ = H i Hi v µ + αh α H α H i= α, H α v µ α Σg + r = µh i µhi v µ + αh α µh α v µ. H i= α, H α α Σg + Hence π v µ = µ, µ + ρ v µ because H α is a non-zero constant multiple of α with the identification a a by, and therefore π v = µ, µ + ρ v because M is generated by v µ. iii Let v π be a lowest weight vector of π. Then we have π v π = π, π ρ v π and therefore π v = π, π ρ v. Similarly we have Θ π v = ϖ, ϖ ρθ v. X α

9 ANNIHILATORS OF GENERALIZED VERMA MODULES 9 Let ϖ be the weight of v. Then we have X α X α X α, X α v = πv Θ π v + ϖ, ρ Θ v α Σg + \Σg Θ Here we note that ϖ, ρ Θ = ϖ, ρ Θ. iv By the same argument as above we have Hence i= = π ϖ, π + ϖ ρ v. X β X β r X β, X β v + H i Hi v βh βh β v = ϖ, ϖ β v. H β, H β X β X β v = ϖ, ϖ β v ϖ + lβ, ϖ + lβ v + β, ϖ + lβ v X β, X β = l ϖ, β + ll β, β v v Suppose π is the adjoint representation of the simple Lie algebra g. Then for H a we have π π H, H = [X α, [X α, H]], H X α, X α α Σg = α Σg = α Σg = H, H. [X α, H], [X α, H] X α, X α αh Hence π π H = H and Bα max, α max + ρ = B α max, α max ρ =. Definition. weights. Let Wπ denote the set of the weights of the finite dimensional irreducible representation π of g. For ϖ Wπ define a real constant.7 D π ϖ = π ϖ, π + ϖ ρ. Here π is the lowest weight of π. Put R + = { α Ψg m αα; m α {,,,...}}. We define a partial order among the elements of Wπ so that ϖ ϖ if and only if ϖ ϖ R +. Moreover we put.8 W Θ π = {ϖ are the highest weights of the irreducible components of π gθ }, W Θ π = {ϖ are the lowest weights of the irreducible components of π gθ }, Wπ aθ = {ϖ aθ ; ϖ Wπ}. Let µ and µ Wπ aθ. Then we define µ Θ µ if and only if µ µ { α Ψg\Θ m αα aθ ; m α {,,,...}}. Remark.. i W π = W π = Wπ and W Θ π = W Θ π. Here π, V denotes the contragredient representation of π, V defined by.9 π Xv v = v πxv for X g, v V and v V. ii Wπ aθ = {ϖ aθ ; ϖ W Θ π} = {ϖ aθ ; ϖ W Θ π}.

10 HIROSHI ODA AND TOSHIO OSHIMA iii Suppose ϖ and ϖ Wπ and put ϖ ϖ = α Ψg m αα. Then ϖ aθ Θ ϖ aθ if and only if m α for any α Ψg \ Θ. Hence π aθ is the smallest element of Wπ aθ. Note that ϖ ϖ if and only if ϖ ϖ. Lemma.. Let ϖ and ϖ Wπ. i If α = ϖ ϖ Ψg, then D π ϖ D π ϖ = ϖ, ϖ ϖ. ii Suppose ϖ W Θ π, ϖ < ϖ and ϖ aθ = ϖ aθ. Then D π ϖ < D π ϖ. Proof. ii Note that D π ϖ D π ϖ = ϖ ϖ, ϖ + ϖ ρ. The assumption in ii implies ϖ ϖ = α Θ m αα with m α. Here at least one of m α is positive. Hence ϖ ϖ, ρ >. Since ϖ are the lowest weights of irreducible representations of g Θ, α, ϖ for α Θ. Thus we have α Θ m αα, ϖ α Θ m αα ρ <. i Put α = ϖ ϖ. Then D π ϖ D π ϖ ϖ, ϖ ϖ = α, ρ α, which equals if α Ψg because w α Σg + \ {α} = Σg + \ {α}. Now we give a key lemma which is used to calculate our minimal polynomial. Lemma.3. Fix an irreducible decomposition κ i= π i, V i of π gθ, V and a basis {v i,,..., v i,mi } of V i so that v i,j are weight vectors for a. Let ϖ i,j and ϖ i be the weight of v i,j and the lowest weight of the representation π i, respectively. Suppose ϖ i,j = ϖ i,j. Then for a positive integer k with k and complex numbers µ,..., µ k k ν= k F π µ ν i,j i,j F π µ ν ν= mod Ugm Θ + n Θ + i,j i,j C s,t,s,t ; ϖ s aθ < Θ ϖ i aθ ϖ s,t=ϖ s,t ϖi µ k + D π ϖ i k F π µ ν ν= s,t s,t. Proof. Note that ϖ i,j ϖ i mod Ugm Θ. It follows from Lemma. that F s,ti,j δ si δ tj ϖ i mod Ugm Θ + n Θ if ϖ s,t ϖ i,j Σg \ Σg Θ.

11 ANNIHILATORS OF GENERALIZED VERMA MODULES Put F l = l ν= F π µ ν. Then Lemma. implies. Fi k,j i,j F k i,j i,j ϖi µ k [F k i,j s,t, F s,ti,j] mod Ugm Θ + n Θ + n Θ Ug = = ϖ s,t ϖ i,j Σg \Σg Θ α Σg + \Σg Θ ϖ s,t =ϖ i,j α α Σg + \Σg Θ ϖ s,t =ϖ i,j α ϖ s,t ϖ s,t =α E s,ti,j, X α [F k X α, X α i,j s,t, X α] E s,ti,j, X α E s,t s,t, X α X α, X α δ ss δ tt F k i,j s,t δ i s δ k j t Fs,t s,t = π ϖ i, π + ϖ i ρ F k i,j i,j α Σg + \Σg Θ ϖ s,t=ϖ s,t =ϖ i,j α E s,ti,j, X α E i,j s,t, X α F k X α, X α s,t s,t. In the above the second equality follows from.8 and Lemma.9 with U = g. The third equality follows from Lemma. with X α = E s,t s,t, X α E s,t s,t ϖ s,t ϖ s,t =α which follows from the identification g gl N together with the property of,. Put X = t X for X MN, C gl N. Let {vi,j } be the dual base of {v i,j} and consider the contragredient representation π of π. Then π X = X for X g with respect to these basis. Then X, Y = X, Y for X, Y g and α Σg + \Σg Θ X αx α X α, X α v i,j = α Σg + \Σg Θ ϖ s,t =ϖ i,j α ϖ s,t =ϖ i,j E s,ts,t, X α E i,js,t, Xα X α, Xα vs,t, which is proved to be equal to D π ϖ i v i,j by Lemma.9 iii because π, ϖ i, ρ for π changes into π, ϖ i, ρ in the dual π with the reversed order of roots. This implies the last equality in.. Note that if D n Θ Ug + Ugm Θ + n Θ satisfies [H, D] = for all H a Θ, then D Ugm Θ + n Θ. Since the condition ϖ i,j ϖ s,t Σg + \ Σg Θ implies ϖ s aθ < Θ ϖ i aθ, we have the lemma. Theorem.4. Retain the notation in Definition.. For ϖ a we identify ϖ aθ with a linear function on a Θ by ϖ a Θ λ = λ Θ, ϖ for λ a Θ. Put. Ω π,θ = { ϖ aθ, D π ϖ ; ϖ W Θ π}, q π,θ x; λ = x µλ C. µ,c Ω π,θ Then q π,θ F π ; λm Θ λ = for any λ a Θ. Proof. For any D Ug there exists a unique constant T D C satisfying T D D mod nug + J Θ λ

12 HIROSHI ODA AND TOSHIO OSHIMA because the dimension of the space M Θ λ/ nm Θ λ equals. Notice that J Θ λ = H a Θ UgH λh + Ugm Θ + n Θ. Use the notation in Lemma.3. Since adhq π,θ F π ; λ i,j i,j = ϖ i,j ϖ i,jhq π,θ F π ; λ i,j i,j for H a, T q π,θ F π ; λ i,j i,j = if ϖ i,j ϖ i,j. Next assume ϖ i,j = ϖ i,j and put Ω π,θ,i = {µ, C Ω π,θ ; µ Θ ϖ i aθ }, q π,θ,i x; λ = x µλ C. µ,c Ω π,θ,i Then qf π i,j i,j J Θ λ for any qx C[x] which is a multiple of q π,θ,i x; λ. It is proved by the induction on ϖ i aθ with the partial order Θ. Take i {,..., κ} so that ϖ i = π. If i = i then Lemma. and Lemma.3 with D π ϖ i = D π π = imply our claim. If i i then π aθ < Θ ϖ i aθ and therefore deg x q π,θ,i x; λ. Hence we can use Lemma.3 again to prove our claim inductively. Thus we get the condition. T q π,θ F π ; λ i,j i,j = for any i, j and i, j. Let Vλ denote the C-subspace of Ug spanned by q π,θ F π ; λ i,j i,j. Then Vλ is adg-stable by Lemma.. The g-module M λ = VλM Θ λ is contained in nm Θ λ because putting u λ = mod J Θ λ, M λ = VλU nu λ = U nvλu λ U n nugu λ = nm Θ λ. On the other hand, since M Θ λ is irreducible if λ belongs to a suitable open subset of a Θ, M λ = {} in the open set. If we fix a base {Y,..., Y m } of n Θ, we have the unique expression q π,θ F π ; λ i,j i,j ν Q ν λy ν Y ν m m mod J Θ λ with polynomial functions Q ν λ. All these Q ν λ vanish on the open set and therefore they are identically zero and we have Vλ J Θ λ for any λ. We have then for any λ M λ = VλUgu λ = UgVλu λ = {}. Theorem.4 is one of our central results since q π,θ x; λ = q π,mθ λx for a generic λ a Θ. Before showing this minimality, which will be done in Theorem.9, we mention the possibility of other approaches to Theorem.4. In fact we have three different proofs. The first one given above has the importance that the calculation in the proof is also used in 3 to study the properties of the two-sided ideal of Ug generated by q π,θ F π ; λ ij. The second one comes from a straight expansion of the method in [Go] and [Go] to construct characteristic polynomials. In the following we first discuss it. The third one is based on infinitesimal Mackey s tensor product theorem which we explain in Appendix A. With this method we shall get the sufficient condition for the minimality of q π,θ x; λ Theorem.9 and slightly strengthen the result of Theorem.4 Theorem.3.

13 ANNIHILATORS OF GENERALIZED VERMA MODULES 3 Definition.5. Let π, V be the contragredient representation of π, V and {v,..., v N } the dual base of the base {v,..., v N } of V. For a g-module M define the homomorphism of associative algebras by.3 h π,m : MN, Ug End M V hπ,m Q N j= u j vj N = i= j= N Qij u j v i for u j M and Q = Q ij MN, Ug. Then QM =, namely, Qij AnnM for any i, j if and only if h π,m Q =. The following lemma is considered in [Go] and [Go]. Lemma.6. Let M be a g-module. For an element N j= u j vj of M V with u j M, we have h π,m F π N u j vj N N = π u j vj + u j π vj N π u j vj. j= j= In particular h π,m F π End g M V. Proof. Let {X,..., X ω } be a base of g and let {X,..., Xω} be its dual base with respect to,. Then N N π u j vj + u j π vj N π u j vj j= j= ν= j= j= ν= j= j= N ω N ω = Xν u j X ν vj X ν u j Xν vj j= ν= i= j= N ω N N = Xν u j X ν, E ij vi + X ν u j Xν, E ij vi = N i= j= N pe ij u j vi. Here we use the fact that Xvj N i= X, E ji v i. i= = N i= X, E ij v i for X g because Xv j = Now we examine the tensor product M V in the preceding lemma when M is realized as a finite dimensional quotient of a generalized Verma module M Θ λ. Proposition.7 a character identity for a tensor product. Put w W χ Λ = sgnwewλ+ρ α Σg +e α e α for Λ a. If Λ, α = for any α Θ, then.4 χ π χ Λ = m π,θϖχ Λ+ϖ by denoting ϖ Wπ m π,θϖ = dim{v V ; Hv = ϖhv H a, Xv = X g Θ n}. Here χ π is the character of the representation π, V and for µ a, e µ denotes the function on a which takes the value e µh at H a.

14 4 HIROSHI ODA AND TOSHIO OSHIMA Proof. It is sufficient to prove.4 under the condition that Λ, α is a sufficiently large real number for any α Ψg \ Θ because both hand sides of.4 are holomorphic with respect to Λ a. Put a = {µ a ; µ, α R α Σg }, a + = {µ a ; µ, α α Σg + }, χ + Λ = w W Θ sgnwe wλ+ρ α Σg e α + e α, w χ ϖ = W Θ sgnw e w ϖ+ρθ α Σg +e α Θ e. α Then χ π = ϖ Wπ m π,θϖ χ ϖ by Weyl s character formula and if ϖ Wπ satisfies m π,θϖ >, then Λ + ϖ a + and χ ϖ χ + Λ e α e α = w W Θ sgnwe wϖ+ρθ α Σg + α Σg Θ e α + e e Λ+ρ α Θ sgnw e w ρθ w W Θ = w W Θ sgnwe wλ+ϖ+ρ e Λ+ϖ+ρ mod µ a \a + Ze µ. For any w W \ W Θ there exists α Σg \ Σg Θ with wα Σg + and then the value wλ + ρ, wα = Λ + ρ, α is sufficiently large and therefore χ ϖ χ Λ χ + Λ e α Ze µ. Hence χ π χ Λ α Σg + e α α Σg + e α e α ϖ W Θ π µ a \a + m π,θϖe Λ+ϖ+ρ mod µ a \a + and we have the proposition because χ π χ Λ α Σg +e α e α is an odd function under W. Lemma.8 eigenvalue. Let π Λ, V Λ be an irreducible finite dimensional representation of g with highest weight Λ. Suppose Λ, α = for α Θ and Λ + ϖ, α for ϖ W Θ π and α Ψg \ Θ. Then the set of the eigenvalues of h π,vλ F π EndV Λ V without counting their multiplicities equals { Λ, ϖ + π ϖ, π + ϖ + ρ ; ϖ W Θ π } = { Λ, ϖ + π ϖ, π + ϖ ρ ; ϖ W Θπ}. Here we identify π with the highest weight of π, V. Proof. The assumption of the lemma and Proposition.7 imply π π Λ = m π,θϖπ Λ+ϖ ϖ W Θπ and hence by Lemma.9 ii and Lemma.6 the eigenvalues of h π,vλ F π are Λ, Λ+ρ + π, π +ρ Λ+ϖ, Λ+ϖ +ρ = Λ, ϖ + π ϖ, π +ϖ +ρ with ϖ W Θ π. Since W Θ π = W Θ π, we have the lemma. Ze µ

15 ANNIHILATORS OF GENERALIZED VERMA MODULES 5 Proof of Theorem.4 the nd version. This proof differs from the previous one in how to deduce the condition.. The rests of two proofs are the same. Note that for fixed i, j and i, j the value T q π,θ F π ; λ i,j i,j depends algebraically on the parameter λ a Θ. Since the set S = {λ a Θ; λ Θ + ϖ, α {,,,...} for ϖ W Θ π {} and α Ψg \ Θ} is Zariski dense in a Θ, we have only to show. for λ S. In this case we have from Lemma.8 and the definition of q π,θ x; λ, h π,vλθ q π,θ F π ; λ = q π,θ h π,vλθ F π ; λ =. Hence q π,θ F π ; λ i,j i,j AnnV λθ for any i, j and i, j. On the other hand, if we take a highest weight vector v λ of V λθ, we get q π,θ F π ; λ i,j i,jv λ T q π,θ F π ; λ i,j i,jv λ + nv λθ and therefore T q π,θ F π ; λ i,j i,j =. Theorem.9 minimality. Let λ a Θ. i The set of the roots of q π,mθλx equals { λ Θ, ϖ + D π ϖ; ϖ W Θ π}. ii If each root of q π,θ x; λ is simple, then q π,θ x; λ = q π,mθλx. Hence we call q π,θ x; λ the global minimal polynomial of the pair π, M Θ λ. Proof. i Fix an irreducible decomposition κ i= U i of the g Θ -module V gθ. Let ϖ i a be the highest weight of U i. With a suitable change of indices we may assume ϖ i aθ < Θ ϖ j aθ implies i > j. Then putting V i = i ν= U ν we get a p Θ -stable filtration {} = V V V κ = V pθ. Note that V i /V i U i is an irreducible p Θ -module on which n Θ acts trivially. Recall M Θ λ M Θ,λΘ = Ug UpΘ U Θ,λΘ and dim U Θ,λΘ =. Hence writing C λ instead of U Θ,λΘ we get by Theorem A. of Appendix A M Θ λ V = Ug UpΘ C λ V Ug UpΘ C λ V pθ. Since C λ C and Ug UpΘ = U n Θ C are exact functors, putting M i = Ug UpΘ C λ V i we get a g-stable filtration with {} = M M M κ = M Θ λ V.5 M i /M i Ug UpΘ C λ U i = M Θ,λΘ+ϖ i. Now as a subalgebra of End M Θ λ V we take A = {D; DM i M i for i =,..., κ}. Then by Lemma.6 and Lemma.9 ii we have h π,mθ λ qf π A for any polynomial qx C[x]. Let η i : A End M i /M i End M Θ,λΘ+ϖ i be a natural algebra homomorphism. Then using Lemma.6 and Lemma.9 ii again we get η i hπ,mθλf π.6 = λ Θ, λ Θ + ρ + π, π + ρ λ Θ + ϖ i, λ Θ + ϖ i + ρ = λ Θ, ϖ i + D π ϖ i. and therefore q π,mθ λ λ Θ, ϖ i + D π ϖ i = q π,mθ λ ηi hπ,mθ λf π = η i hπ,mθλ qπ,mθλf π =.

16 6 HIROSHI ODA AND TOSHIO OSHIMA Since {ϖ i } = W Θ π = W Θ π we can conclude λ Θ, ϖ + D π ϖ is a root of the minimal polynomial for each ϖ W Θ π. Conversely Theorem.4 assures any other roots do not exist. ii The claim immediately follows from i and the definition of q π,θ x; λ. Remark.3. In general it may happen for a certain λ that q π,θ x; λ q π,mθ λx. Such example is shown in [O4] when g is o n and λ is invariant under an outer automorphism of g, which is related to the following theorem. It gives more precise information on our minimal polynomials. Theorem.3. Let λ a Θ. Let W Θπ = W λ W λ W m λ λ be a division of W Θ π into non-empty subsets W l λ such that the relation λ Θ ϖ {w.λ Θ ϖ ; w W } holds for ϖ, ϖ W Θ π if and only if ϖ, ϖ W l λ for some l. For each l we denote by κ l the maximal length of sequences {ϖ, ϖ,..., ϖ } of weights in W l λ such that the restriction of each weight to a Θ gives both strictly and linearly ordered sequences: ϖ aθ < Θ ϖ aθ < Θ < Θ ϖ aθ. i λ Θ, ϖ + D π ϖ = λ Θ, ϖ + D π ϖ if ϖ, ϖ W l λ for some l. ii Let qx C[x] and suppose for each l =,..., m λ, qx is a multiple of x λ Θ, ϖ D π ϖ κ l with ϖ W l λ. Then qf π M Θ λ =. Proof. i By the W -invariance of, and the assumption, we have λ Θ + ρ ϖ, λ Θ + ρ ϖ = λ Θ + ρ ϖ, λ Θ + ρ ϖ, which implies the claim. ii Use the notation in the proof of Theorem.9. Let M be a g-module and µ a. We say that a non-zero vector v in M is a generalized weight vector for the generalized infinitesimal character µ if for any Zg there exists a positive integer k such that a µ k v =. We denote by M µ the submodule of M spanned by the generalized weight vectors for the generalized infinitesimal character µ. Note that M µ = M µ if and only if µ = w.µ for some w W. By virtue of.5 and Remark.6, M Θ λ V is uniquely decomposed as a direct sum of submodules in {M Θ λ V λθ +ϖ ν ; ν =,..., i}. For i =,..., κ using a p Θ -module V [i] = U i U ν V i, ν; ϖ i aθ < Θ ϖ ν aθ define M [i] = Ug pθ C λ V [i] = U n Θ C λ V [i]. It is naturally considered as a g-submodule of M i = U n Θ C λ V i. If we define the surjective homomorphism then.7 Ker τ [i] = τ [i] : M [i] M i M i /M i M Θ,λΘ +ϖ i, ν; ϖ i aθ < Θϖ ν aθ M [ν]. Since M Θ,λΘ +ϖ i has infinitesimal character λ Θ + ϖ i we get M [i] = M [i] λθ +ϖ i + M [ν]. ν; ϖ i aθ < Θϖ ν aθ

17 ANNIHILATORS OF GENERALIZED VERMA MODULES 7 Therefore we get inductively.8 M [i] = M [i] λθ+ϖ i + ν; ϖ i aθ < Θϖ ν aθ M [ν] λθ+ϖ ν. Notice that the g-homomorphism h π,mθ λf π leaves any g-submodule of M Θ λ V stable. Then from.6 and.7 h π,mθλf π λ Θ, ϖ i D π ϖ i M [i] λθ+ϖ i = = ν; ϖ i aθ < Θ ϖ ν aθ M [ν] ν; ϖ i aθ < Θϖ ν aθ λ Θ +ϖ i M [ν] λθ +ϖ ν ν; ϖ i aθ < Θϖ ν aθ, λ Θ +ϖ ν {w.λ Θ +ϖ i ; w W } λ Θ+ϖ i M [ν] λθ +ϖ ν. By the relation {ϖ i } = W Θ π = W Θ π and the assumption of ii we get inductively h π,mθλqf π M [i] λθ+ϖ i = qh π,mθλf π M [i] λθ+ϖ i = {} for i =,..., κ. Now our claim is clear because by.8 we have κ κ M Θ λ V = M [i] = M [i] λθ +ϖ i. i= Corollary.3. Let τ be an involutive automorphism of g which corresponds to an automorphism of the Dynkin diagram of g. Then τa = a and τn = n. Furthermore we suppose τp Θ = p Θ, or equivalently, τa Θ = a Θ. For ϖ a we identify ϖ aθ τ as a linear function on a Θ τ by ϖ aθ τ λ = λ Θ, ϖ for λ a Θ τ. Put i= Ω π,θ,τ = { ϖ aθ τ, D πϖ ; ϖ W Θ π}, q π,θ,τ x; λ = x µλ C. µ,c Ω π,θ,τ Then for λ a Θ τ we have the following. i q π,θ,τ F π ; λm Θ λ =. ii If each root of q π,θ,τ x; λ is simple, then q π,θ,τ x; λ = q π,mθ λx. Proof. We naturally identify ρ Θ with an element in a Θ τ. For a given pair of weights ϖ, ϖ W Θ π with ϖ aθ < Θ ϖ aθ, choose the non-negative integers {m α ; α Ψg\Θ} so that ϖ aθ ϖ aθ = α Ψg\Θ m αα aθ. Then ϖ aθ ρ Θ ϖ aθ ρ Θ = α Ψg\Θ m α α, ρ Θ >. It simply shows ϖ aθ τ, D πϖ ϖ aθ τ, D πϖ. Hence from Theorem.3 we get i. Now ii is clear from Theorem.9. We will shift a by ρ so that the action w.µ = wµ + ρ ρ for µ a and w W changes into the natural action of W and then we can give the characteristic polynomial as a special case of the global minimal polynomials. The result itself is not new and it has already been studied in [Go].

18 8 HIROSHI ODA AND TOSHIO OSHIMA Theorem.33 Cayley-Hamilton [Go]. The characteristic polynomial q π x of π is given by.9 q π x = π, π + ρ ϖ, ϖ x ϖ ϖ Wπ under the identification C[x] Sa W C[x] Sa W Zg[x] by the symmetric bilinear form, and the Harish-Chandra isomorphism: Zg Ua W ; Υ, Υ µ = a µ ρ for µ a. Here π is identified with its highest weight. In particular q π x Zg[x]. Proof. Note that π, π + ρ = π, π ρ. Let q π x be the element of Zg[x] identified with the right-hand side of.9. Put V = i, j C q πf π ij and V a = {D a ; D V}. Then Theorem.4 with Θ = shows Qµ = for any µ a and Q V a, which implies V a = {}. Since V is adg-stable, we have V = {} as is shown in [O, Lemma.]. Since the minimality of q π x follows from Theorem.9, we get q π x = q π x. Corollary.34. i Let g be a simple Lie algebra. Then the characteristic polynomial of the adjoint representation of g is given by Bα, α q αmax x = x α. α Σg {} Here B, denotes the Killing form of g. ii Suppose that the representation π is minuscule, that is, Wπ is a single W -orbit. Then q π x = x ϖ π, ρ. ϖ Wπ Proof. This is a direct consequence of Theorem.33 and Lemma.9 v. Corollary.35. Put q π x = x m + x m + + m x + m with j Zg and define F π = Fπ m Fπ m m I N. Then F π Fπ = F π F π = m I N = ϖ Wπ ϖ π, π + ρ ϖ, ϖ I N, In particular, F π is invertible in M N, Ẑg Zg Ug with the quotient field Ẑg of Zg. In the next definition and the subsequent proposition, we do not assume.7. Namely, g is a general reductive Lie algebra and π, V denotes a finite dimensional irreducible representation which is not necessarily faithful. Moreover we use the symbol, for the symmetric bilinear form on a defined by the restriction of the Killing form of g. Definition.36 dominant minuscule weight. We say a weight π min of π is dominant and minuscule if and π min, α for all α Σg + π min, π min ϖ, ϖ for all ϖ Wπ.

19 ANNIHILATORS OF GENERALIZED VERMA MODULES 9 If the highest weight of π is dominant and minuscule, then π, V is called a minuscule representation. α α,α Proposition.37. Put Ψg = {α,..., α r } and define α = for α Σg. Let π, V be a finite dimensional irreducible representation of g. Let π min be a dominant minuscule weight of π. i If the highest weight of π is in the root lattice, then π min =. ii π min is uniquely determined by π. Moreover if π, V is a finite dimensional irreducible representation of g such that the difference of the highest weight of π and that of π is in the root lattice of Σg, then π min = π min. iii ϖ Wπ is a dominant minuscule weight if and only if.3 ϖ, α {, } for all α Σg +. iv If π is a minuscule representation, then Wπ = W π min. v Suppose g is simple. Let Σg := {α ; α Σg} be the dual root system of Σg. Let β be the maximal root of Σg and put β = r i= n iαi. Define the fundamental weights Λ i by Λ i, αj = δ ij. Then π is a minuscule representation if and only if its highest weight is or Λ i with n i =. Proof. For α Σg we denote by g α the Lie algebra generated by the root vectors corresponding to α and α. Note that g α is isomorphic to sl. i Suppose the highest weight of π is in the root lattice. Put ϖ = r i= m iϖα i for ϖ Wπ. Note that m i ϖ are integers. Let ϖ W π such that m i ϖ and r i= m iϖ r i= m iϖ for ϖ W π satisfying m i ϖ for i =,..., r. The existence of ϖ is clear because m i π for i =,..., r. Suppose ϖ. Since < ϖ, ϖ = r i= m iϖ ϖ, α i, there exists an index k such that ϖ, α k > and m k ϖ >. Hence ϖ α k Wπ by the representation π g α k, which contradicts the assumption for ϖ. Thus = ϖ Wπ and π min =. ii iv Suppose the existence of α Σg + with π min, α >. Then it follows from the representation π g α that π min α Wπ and π min, π min π min α, π min α = π min, α α, α >, which contradicts the assumption of π min. Thus we have.3 for ϖ = π min. Suppose π is an irreducible representation of g with the highest weight ϖ satisfying.3. Suppose Wπ W ϖ. Then there exist µ W ϖ and µ Wπ such that µ / W ϖ with α := µ µ Σg. By the W -invariance we may assume µ = ϖ and therefore µ = ϖ α with α Σg +. Then by the representation π g α together with the condition.3 we have ϖ, α = and µ = w α ϖ, which is a contradiction. Thus we have iv. Let ϖ and ϖ be the elements of a satisfying the condition.3. Then ϖ := ϖ ϖ satisfies ϖ, α {,, } for α Σg. Suppose that ϖ is in the root lattice. Let ϖ W ϖ such that ϖ, α for α Σg +. Since ϖ also satisfies.3, the finite dimensional irreducible representation π with the highest weight ϖ is minuscule by the argument above. Since ϖ is in the root lattice, ϖ = by i and hence ϖ = ϖ. Thus we obtain ii and iii. v Let α Σg +. If we denote α = r i= n iααi, then n iα n i for i =,..., r. Hence the claim is clear. Remark.38. Equivalent contents of Proposition.37 are found in exercises of [Bo], Ch. VI. Restore the previous setting.7 on g and π, V. Proposition.39. i Let V ϖ denote the weight space of V with weight ϖ Wπ. Define the projection map p Θ : Wπ Wπ aθ by p Θ ϖ = ϖ aθ and put V Λ =

20 HIROSHI ODA AND TOSHIO OSHIMA ϖ p Θ Λ V ϖ for Λ Wπ aθ. Then.3 V = V Λ Λ Wπ aθ is a direct sum decomposition of the g Θ -module V. Let V Λ = V Λ V Λ kλ be a decomposition into irreducible g Θ -modules. We denote by ϖ Λ the dominant minuscule weight of π gθ, V Λ. Then.3 V ϖλ = k Λ i= V ϖλ V Λ i with dim V ϖλ V Λ i >. In particular, V Λ is an irreducible g Θ -module if dim V ϖλ =. ii Put Ψg = {α,, α r } and put Ψg \ Θ = {α i,..., α ik }, define the map and put for m L Θ. Then p Θ : Σg Z k α = m i α i m i,..., m ik L Θ = {} {p Θ α; α Σg}, CX α if m, α p V m = Θ m a + CX α if m = α p Θ m.33 g = m L Θ V m is a decomposition of the g Θ -module g. If m, then V m is an irreducible g Θ - module. On the other hand, V = g Θ is isomorphic to the adjoint representation of g Θ = a Θ m Θ. Let Θ = Θ Θ Θ l be the division of Θ into the connected parts of vertexes in the Dynkin diagram of Ψg. Then m Θ = m Θ m Θ m Θl gives a decomposition into irreducible g Θ -modules. iii Suppose that the representation π, V is minuscule. Put W π = {w W ; wπ = π}. Here we identify π with its highest weight. Let {w,..., w k } be a representative system of W π \W/W Θ such that w i W Θ. Then with the notation in i.34 V = k i= V w i π aθ gives a decomposition into irreducible g Θ -modules. V w i π aθ has highest weight w i π. Moreover the g Θ -submodule Proof. i Since α aθ = for α Θ,.3 is a decomposition into g Θ -modules. Then Proposition.37 ii implies that ϖ Λ is the minuscule weight for any π gθ, V Λ i and therefore the other statements in i are clear. ii Note that α ik aθ,..., α i aθ are linearly independent and g Θ = V. Then the statements in ii follows from i. iii From i each V w i π aθ is an irreducible g Θ -module and Since w i W Θ we have w i weight of V w i V w i π aθ {V w π; w W π w i W Θ }. π aθ is w i π + α / Wπ for α Σg Θ +. It shows the highest π. Since w i π w j π if i j we have.34.

21 ANNIHILATORS OF GENERALIZED VERMA MODULES We give the minimal polynomials for some representations in the following proposition as a corollary of Lemma.9 v and Proposition.39. Proposition.4. Retain the notation in Theorem.4 and Proposition.39. i multiplicity free representation Suppose dim V ϖ = for any ϖ Wπ. Let Λ be the lowest weight of π gθ, V Λ for Λ Wπ aθ. Then q π,θ x; λ = x λ Θ, Λ π Λ, π + Λ ρ.35 = Λ Wπ aθ Λ Wπ aθ x λ Θ + ρ, Λ + π, ρ π, π Λ, Λ. ii adjoint representation Suppose g is simple and Θ. Let Θ = Θ Θ l be the division in Proposition.39 ii. Let αmax i denote the maximal root of the simple Lie algebra m Θi for i =,..., l. Put Ω Θ = {Bα max, α max + ρθ,..., Bα l max, α l max + ρθ l }. Let α m be the smallest root in p Θ m for m L Θ \ {} under the order in Definition.. Then for the adjoint representation of g,.36 q αmax,θx; λ = x C Ω Θ m L Θ\{} x C x Bλ Θ + ρ, α m Bα m, α m iii minuscule representation Suppose π, V is minuscule. Then with w,..., w k in Proposition.39 iii, k.37 q π,θ x; λ = x w i λθ + ρ Θ ρθ + ρ, π. i= Proof. It is easy to get i and ii. iii Let w Θ denote the longest element in W Θ. Then the g Θ -module V w i π aθ has lowest weight w Θ w i π. The claim follows from the next calculation: λ Θ, w Θ w i π + π w Θw i π, π + w Θ w i π ρ. = λ Θ + ρ, w Θ w i π + ρ, π = w i w Θ λ Θ + ρ + ρ, π = w i λθ + ρ Θ ρθ + ρ, π. 3. Two-sided ideals Our main concern in this paper is the following two-sided ideal. Definition 3. gap. Let λ a Θ. If a two-sided ideal I Θλ of Ug satisfies 3. J Θ λ = I Θ λ + Jλ Θ, then we say that I Θ λ describes the gap between the generalized Verma module M Θ λ and the Verma module Mλ Θ. It is clear that there exists a two-sided ideal I Θ λ satisfying 3. if and only if 3. J Θ λ = Ann M Θ λ + Jλ Θ. This condition depends on λ but such an ideal exists and is essentially unique for a generic λ cf. Proposition 3., Theorem 3., Remark 4.4. The main purpose in this paper is to construct a good generator system of the ideal from a minimal polynomial.

22 HIROSHI ODA AND TOSHIO OSHIMA Definition 3. two-sided ideal. Using the global minimal polynomial defined in the last section, we define a two-sided ideal of Ug: 3.3 I π,θ λ = Ugq π,θ F π ; λ ij + Ug a λ Θ. i,j Zg From Theorem.4 and Remark.6 this ideal satisfies 3.4 I π,θ λ J Θ λ. In this section we will examine the condition so that 3.5 J Θ λ = I π,θ λ + Jλ Θ. Proposition 3.3 invariant differential operators. For Zg and a nonnegative integer k we denote by k a the homogeneous part of a with degree k and put 3.6 T π k = m π ϖϖ k. ϖ Wπ Here m π ϖ is the multiplicity of the weight ϖ of π and we use the identification ϖ a a Ua. Let {,..., r } be a system of generators of Zg as an algebra over C and let d i be the degree of i a for i =,..., r. We assume that d a,..., r d r a are algebraically independent. Suppose a subset A of {,..., r} satisfies { d k deg 3.7 x q π,θ x, λ if k {,..., r} \ A, C[ d a,..., r d r a ] = C[ i d i a, T d k π ; i A, k {,..., r} \ A]. Then 3.8 I π,θ λ = i,j Ugq π,θ F π ; λ ij + i A Ug i i a λ Θ. Proof. Note that i,j Ugq π,θf π ; λ ij Trace Fπ ν q π,θ F π ; λ if ν. On the other hand, since Trace F l k π q π,θ Fπ ; λ d k = T d k a π by Lemma.3 with Θ = if the integer l k = d k deg x qπ,θ F π ; λ is non-negative, the assumption implies that for k A, k may be replaced by Trace F l k π q π,θ Fπ ; λ, which implies the proposition. Lemma 3.4. Let V be an adg-stable subspace of Ug and let V = ϖ V ϖ be the decomposition of V into the weight spaces V ϖ with weight ϖ a. Suppose D a λ Θ = for D V. Then the following three conditions are equivalent. i J Θ λ UgV + Jλ Θ. ii For any α Θ there exists D V α such that D X α Jλ Θ. iii For any α Θ there exists D V such that D a λ Θ α. Proof. Let Ug = ϖ Ug ϖ be the decomposition of Ug into the weight spaces Ug ϖ with weight ϖ a. Let µ a. Since Ug = U n Jµ, to D Ug, there corresponds a unique D µ U n such that D D µ Jµ. Here we note that D Ug ϖ implies D µ U n ϖ and that D µ = D a µ C whenever D Ug. Put V µ = {D µ ; D V}. Since adxv V for X b, we have P D V + Jµ and therefore P D µ V µ for every P Ub and D V. Owing to Ug = U n Ub, we have 3.9 {D µ ; D UgV} = U nv µ. Note that 3. V µ = {V ϖ µ ; ϖ = γ Ψg n γ γ for some non-negative integers n γ }.

23 ANNIHILATORS OF GENERALIZED VERMA MODULES 3 Suppose i. Let α Θ. Since X α J Θ λ \ Jλ Θ, there exists D UgV with D λθ = X α. On the other hand, we can deduce U nv α λθ = V α λθ from 3. because the assumption of the lemma assures V λθ =. Hence from 3.9 we may assume D V α. Thus we have ii. It is clear that ii implies i because J Θ λ = Jλ Θ + α Θ UgX α. Let α Θ. Since adhx α = αhx α for H a, we have H H k X α = X α H αh H k αh k for H,..., H k a. We also have X γ X α Jλ Θ for γ Σg + because λ Θ [X α, X α ] = and [X γ, X α ] n if γ α. Hence for any D Ug, 3. adx α D λθ = [X α, D a ] λθ = D a λ Θ D a λ Θ α X α. Now it is clear that iii implies ii. Conversely suppose ii. Let α Θ. Since V α = adx α V, there exists D V with adx α D λ Θ = X α and we have iii from 3.. Remark 3.5. In the above lemma λ Θ α = w α.λ Θ for α Θ because λ Θ, α =. By the Duflo theorem [Du], Ann Mµ = Zg Ug a µ for any µ a. Then, by the following theorem, each Ann Mµ has the same adg- module structure. Theorem 3.6 the Kostant theorem [Ko]. There exists an adg-submodule H of Ug such that Ug is naturally isomorphic to Zg H by the multiplication. For any finite dimensional g-module V, dim Hom g V, H = dim V. Similarly on the annihilators of generalized Verma modules we have Proposition 3.7. Suppose λ Θ + ρ is dominant. Then for any finite dimensional g-module V and H in Theorem 3.6, dim Hom g V, Ann MΘ λ / Ann Mλ Θ where V gθ = {v V; Xv = X g Θ }. = dim Hom g V, H Ann MΘ λ = dim V dim V g Θ Before proving the proposition, we accumulate some necessary facts from [BGG], [BG] and [J]. Definition 3.8 category O [BGG]. Let O be the abelian category consisting of the g-modules which are finitely generated, a -diagonalizable and Un-finite. All subquotients of Verma modules are objects of O. For µ a we denote by Lµ the unique irreducible quotient of the Verma module Mµ. There exists a unique indecomposable projective object P µ O such that Hom g P µ, Lµ. Proposition 3.9 [BGG], [BG]. i If µ + ρ is dominant, then P µ = Mµ and { dim Hom g Mµ, Mµ if µ = µ, = if µ µ. ii For any µ, µ a dim Hom g P µ, Lµ = { if µ = µ, if µ µ. iii For any finite dimensional g-module V and µ a, V P µ is a projective object in O.

24 4 HIROSHI ODA AND TOSHIO OSHIMA Proposition 3. [BG], [J]. Suppose µ a and µ + ρ is dominant. Then the map 3. {I Ug; two-sided ideal, I Ann Mµ } {M Mµ; submodule} defined by I IMµ is injective and hence Ann Mµ/IMµ = I for any twosided ideal I with I Ann Mµ. The image of the map 3. consists of the submodules which are isomorphic to quotients of direct sums of P µ with 3.3 µ + ρ, β β, β {,,,...} for any β Σg + such that µ + ρ, β =. Proof of Proposition 3.7. We first show the map 3.4 Hom g V, H ϕ Φ Hom g V Mλ Θ, Mλ Θ defined by Φv u = ϕvu is a linear isomorphism. Since Ug = H Ann Mλ Θ the map is injective. To show the surjectivity we calculate the dimensions of both spaces. By Theorem 3.6 dim Hom g V, H = dim V. On the other hand, note that Hom g V Mλ Θ, Mλ Θ Hom g Mλ Θ, Mλ Θ V and there exist a sequence {µ,..., µ l } a and a g-stable filtration {} = M M M l = Mλ Θ V such that M i /M i Mµ i for i =,..., l. Here the number of appearances of λ Θ in the sequence {µ,..., µ l } equals dim V = dim V cf. the proof of Theorem.9. Since λ Θ + ρ is dominant, it follows from Proposition 3.9 i that dim Hom g Mλ Θ, Mλ Θ V = dim V. Thus 3.4 is isomorphism. Secondly, consider the exact sequence J Θ λ/jλ Θ Mλ Θ M Θ λ. It is clear that under the isomorphism 3.4 the subspace corresponds to the subspace Hom g V, H Ann MΘ λ Hom g V, H Hom g V Mλ Θ, J Θ λ/jλ Θ Hom g V Mλ Θ, Mλ Θ. Let us calculate the dimension of the latter space. By Proposition 3.9 i and iii, V Mλ Θ is projective and therefore dim Hom g V Mλ Θ, J Θ λ/jλ Θ Here we know = dim Hom g V Mλ Θ, Mλ Θ dim Hom g V Mλ Θ, M Θ λ. Hom g V Mλ Θ, M Θ λ Hom g Mλ Θ, M Θ λ V and there exist a sequence {µ,..., µ l } a and a g-stable filtration {} = M M M l = M Θ λ V such that M i /M i M Θ,µi for i =,..., l. The number of appearances of λ Θ in the sequence {µ,..., µ l } equals dimv g Θ = dim V g Θ cf. the proof of Theorem.9. Since the generalized Verma module M Θ,µi is a quotient of Mµ i, Proposition 3.9 i implies dim Hom g Mλ Θ, M Θ λ V = dim V gθ. Thus the proposition is proved.